Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

80 PLANETARY TIIEORY.
value only commensurate with their value in r$r, and consequently, though considerable, they cannot rise to first-rate
importance. We shall therefore confine our attention to the
term depending on cos (pnt - qn't + Q) where pn - qn' is
very small. The period of the force which produces this
27r
terni is n-, and is therefore very great. The period
pn - qn
of the inequality is the same. It appears then that the
terms which l)ecome very great are those which occupy a
very long period, and which in consequence can hardly be
discovered from observations that extend through a short
time. Thus the inequality of Saturn produced by Jupiter
(and a corresponding one of Jupiter) depending on
cos5(nt + ) -2(n't + ) + Q,
has a period exceeding 900 years, or 30 revolutions of the
exterior planet. The inequality of the Earth alluded to
in (94) las a period of about 240 years, or 240 revolutions
of the exterior planet. There are many similar terms in
the theory of the different planets, but no others so remarkable as these.
101. The increase of the coefficient from integration is
inversely as (p z - qn')2, or directly as ( ----, or directly
\pBn-q/n
as thi square of the period of the inequality. Of this, the
following popular explanation may be offered. If we conceive a force to urge the planet in the same direction for
a long time, or if we conceive the preponderance of forces
acting in one direction over those acting in the opposite
direction to be of the same kind for a long time, (which
may be the case when pn - qn' is small, because then the
periodic times of the two planets are in the proportion of
p: q very nearly, and therefore after q revolutions of one
planet or p revolutions of the other, they will again be in
nearly the saine relative situation at the same parts of their
orbits, and thus the actions of the same kind are repeated
over and over again for a long time), the velocity of the
planet may be increased so that it describes a larger orbit
than before, and its periodic time will therefore be increased,